where \(\mathbf {\mu }_{k} \in \mathbb {R}^{d_z}\)  is the prior mean vector and \(\Sigma _k \in \mathbb {S}_{++}^{d_z}\)  is the positive definite prior covariance matrix. The observed data consists of count vectors of the transcriptomes, which are modeled as multinomial distributed [1]}. We model the conditional distributions of the observed count vectors of species \(l\)  as follows:
\( p(\mathbf {x}_{kl,i} | \mathbf {z}_{k,i}, \Theta _{kl}) = Mu(\mathbf {x}_{kl,i} | N_{kl,i}, \mathcal {S}(\Theta _{kl}^T \mathbf {z}_{k,i}),\) 
